Fractals: the same pattern at every scale

Self-similar shapes

Look closely at a fern. Each leaf is a small copy of the whole fern, and each part of a leaf is a smaller copy again. A tree trunk divides into limbs, limbs into branches and branches into twigs; clouds, coastlines, mountains and lightning are built the same way. Shapes like these, which repeat their own pattern at ever smaller sizes, are called fractals. Mathematics can build fractals step by step with very simple rules.

The Sierpinski carpet

The Polish mathematician Sierpinski took a square, cut it into 9 equal squares and removed the centre one. Then he did the same to each of the 8 squares left, and so on. Let Rₙ be the squares remaining and Hₙ the holes after step n. Every remaining square gives 8 squares at the next step, so Rₙ₊₁ = 8Rₙ and Rₙ = 8ⁿ. Every remaining square also makes one new hole, while old holes stay, so Hₙ₊₁ = Hₙ + Rₙ: the holes go 0, 1, 9, 73, …, that is 1 + 8 + 8² + … + 8ⁿ⁻¹. Since 8/9 of the area survives each step, the area left after n steps is (8/9)ⁿ of the first square.

The Sierpinski triangle

The same idea works with an equilateral triangle. Joining the midpoints of its sides splits it into 4 identical equilateral triangles, because each corner triangle has two equal halves of sides around a 60° angle and so is itself equilateral. Remove the middle triangle, then repeat on the 3 that remain. After n steps there are 3ⁿ triangles, 1 + 3 + … + 3ⁿ⁻¹ holes, and the area left is (3/4)ⁿ.

The Koch snowflake

Described by the Swedish mathematician von Koch in 1904, this one also starts with an equilateral triangle. Divide each side into three equal parts, raise an equilateral triangle on the middle part and remove that middle part, so each side becomes a bump of 4 shorter sides. Repeat on every side of the new shape. Starting with sides of 1 unit, step n has 3 × 4ⁿ sides, each (1/3)ⁿ long, so the perimeter is 3 × (4/3)ⁿ: 3, 4, 16/3, 64/9, … It keeps growing with every step.

Fractals in art

Perhaps the oldest human-made fractals are in India’s temples. The Kandariya Mahadev Temple at Khajuraho, finished around 1025 CE, rises in a tower made of smaller copies of the full tower, carrying still smaller copies, and similar patterns appear in temples in Madurai, Hampi, Rameswaram and Varanasi. Nigerian Fulani wedding blankets nest diamond patterns inside diamonds, and the Dutch artist M.C. Escher’s print ‘Smaller and Smaller’ repeats lizards at shrinking sizes.

Why fractals matter

Fractals are more than pretty pictures. Their exact counts follow from simple rules, which makes them a good way to practise powers and patterns: 8ⁿ, 3ⁿ and 4ⁿ grow very fast, while (8/9)ⁿ and (3/4)ⁿ shrink towards zero. They also behave strangely. The Sierpinski carpet ends up with almost no area but countless holes, and the Koch snowflake stays inside a fixed circle while its boundary grows longer than any length you choose. Shapes like these help scientists describe rough, branching things such as coastlines, blood vessels and river networks.

More from this chapter: visualising solids, nets of solids, the chapter notes and the NCERT solutions.